Mobile games are crazy expensive when falling for the in-app purchases trap. It is a trap, in my opinion; even more so when it isn’t debited to the player’s own payment method.
[JPY 162000 ≈ USD 1086 ≈ EUR 1038]
The support page has instruction on how to request refunds: Request a refund for apps or content that you bought from Apple - Apple Support. Do so for each transaction that applies. I think there will be a comment field where you can state ‘why’ concisely. ‘Child’, ‘without consent’ and ‘unintended’ from the family’s point of view, seem like keywords that could get included, if I understand you correctly. Copy-paste your comment for each one; repetition is okay. I would advise to do all requests in one session without pauses, then the staff can apply the same verdict for all transactions the same.
You should also take action to prevent repetition. If the staff agrees once, then they may not do so a next time.
Approve what kids buy with ‘Ask to Buy’ - Apple Support
Or not let the player have access to the family’s devices or payment methods.